Admissible Invariants of genus 3 Curves

TL;DR

This paper computes specific invariants for all genus 3 curves, providing sharp lower bounds that enhance understanding in arithmetic geometry and support the Effective Bogomolov Conjecture.

Contribution

It explicitly calculates admissible invariants for genus 3 curves and establishes sharp lower bounds, improving previous bounds related to the Effective Bogomolov Conjecture.

Findings

Explicit formulas for invariants of genus 3 curves.

Sharp lower bounds for invariants , , and .

Enhanced bounds support the Effective Bogomolov Conjecture.

Abstract

Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we explicitly calculated these admissible invariants for all curves of genus $3$. We find the sharp lower bound for the invariants $\varphi$, $\lambda$ and $\epsilon$ for all polarized metrized graphs of genus $3$. This improves the lower bound given for Effective Bogomolov Conjecture for such curves.